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Monday, October 29, 2018 - 13:55 ,
Location: Skiles 005 ,
Prof. Tobin Issac ,
Georgia Tech, School of Computational Science and Engineering ,
Organizer: Sung Ha Kang

We are often forced to make important decisions with imperfect and incomplete data. In model-based inference, our efforts to extract useful information from data are aided by models of what occurs where we have no observations: examples range from climate prediction to patient-specific medicine. In many cases, these models can take the form of systems of PDEs with critical-yet-unknown parameter fields, such as initial conditions or material coefficients of heterogeneous media. A concrete example that I will present is to make predictions about the Antarctic ice sheet from satellite observations, when we model the ice sheet using a system of nonlinear Stokes equations with a Robin-type boundary condition, governed by a critical, spatially varying coefficient. This talk will present three aspects of the computational stack used to efficiently estimate statistics for this kind of inference problem. At the top is an posterior-distribution approximation for Bayesian inference, that combines Laplace's method with randomized calculations to compute an optimal low-rank representation. Below that, the performance of this approach to inference is highly dependent on the efficient and scalable solution of the underlying model equation, and its first- and second- adjoint equations. A high-level description of a problem (in this case, a nonlinear Stokes boundary value problem) may suggest an approach to designing an optimal solver, but this is just the jumping-off point: differences in geometry, boundary conditions, and otherconsiderations will significantly affect performance. I will discuss how the peculiarities of the ice sheet dynamics problem lead to the development of an anisotropic multigrid method (available as a plugin to the PETSc library for scientific computing) that improves on standard approaches.At the bottom, to increase the accuracy per degree of freedom of discretized PDEs, I develop adaptive mesh refinement (AMR) techniques for large-scale problems. I will present my algorithmic contributions to the p4est library for parallel AMR that enable it to scale to concurrencies of O(10^6), as well as recent work commoditizing AMR techniques in PETSc.

Series: Geometry Topology Seminar

Series: Geometry Topology Seminar

Series: Research Horizons Seminar

In this chalk plus slides talk, I will give a few examples from my own experience to illustrate how one can use stochastic differential equations in various applications, and its theoretical connection to diffusion theory and optimal transport theory. The presentation is designed for first or second year graduate students.

Series: High Dimensional Seminar

Alesker has introduced the notion of a smooth valuation on a smooth manifold M. This is a special kind of set function, defined on sufficiently regular compact subsets A of M, extending the corresponding idea from classical convexity theory. Formally, a smooth valuation is a kind of curvature integral; informally, it is a sum of Euler characteristics of intersections of A with a collection of objects B. Smooth valuations admit a natural multiplication, again due to Alesker. I will aim to explain the rather abstruse formal definition of this multiplication, and its relation to the ridiculously simple informal counterpart given by intersections of the objects B.

Series: Analysis Seminar

The centerpiece of the subject of integral geometry, as conceived originally by Blaschke in the 1930s, is the principal kinematic formula (PKF). In rough terms, this expresses the average Euler characteristic of two objects A, B in general position in Euclidean space in terms of their individual curvature integrals. One of the interesting features of the PKF is that it makes sense even if A and B are not smooth enough to admit curvatures in the classical sense. I will describe the state of our understanding of the regularity needed to make it all work, and state some conjectures that would extend it.

Series: Graph Theory Working Seminar

A
graph G is H-free if H is not isomorphic to an induced subgraph of G.
Let Pt denote the path on t vertices, and let Kn denote the complete
graph on n vertices. For a positive integer r, we use rG to denote the
disjoint
union of r copies of G. In this talk, we will discuss the result, by
Gaspers and Huang, that (2P2, K4)-free graphs are 4-colorable, where the
bound is attained by the five-wheel and the complement of seven-cycle.
It answers an open question by Wagon in 1980s.

Series: Graph Theory Seminar

Let X denote the number of triangles in the random graph G(n, p). The problem of determining the asymptotics of the rate of the upper tail of X, that is, the function f_c(n,p) = log Pr(X > (1+c)E[X]), has attracted considerable attention of both the combinatorics and the probability communities. We shall present a proof of the fact that whenever log(n)/n << p << 1, then f_c(n,p) = (r(c)+o(1)) n^2 p^2 log(p) for an explicit function r(c). This is joint work with Matan Harel and Frank Mousset.

Series: ACO Student Seminar

Abstract In
this talk, I will present a popular distributed method, namely,
distributed consensus-based gradient (DCG) method, for solving optimal
learning problems over a network of agents. Such problems arise in many
applications such as, finding optimal parameters over
a large dataset distributed among a network of processors or seeking an
optimal policy for coverage control problems in robotic networks. The
focus is to present our recent results, where we study the performance
of DCG when the agents are only allowed to exchange
their quantized values due to their finite communication bandwidth. In
particular, we develop a novel quantization method, which we refer to as
adaptive quantization. The main idea of our approach is to quantize the
nodes' estimates based on the progress of
the algorithm, which helps to eliminate the quantized errors. Under
the adaptive quantization, we then derive the bounds on the convergence
rates of the proposed method as a function of the bandwidths
and the underlying network topology, for both convex and strongly convex
objective functions. Our results suggest that under the adaptive
quantization, the rate of convergence of DCG with and without
quantization are the same, except for a factor which captures
the number of quantization bits. To the best of the authors’ knowledge,
the results in this paper are considered better than any existing
results for DCG under quantization.
This is based on a joint work with Siva Theja Maguluri and Justin Romberg.
Bio Thinh
T. Doan is a TRIAD postdoctoral fellow at Georgia Institute of
Technology, joint between the School of Industrial and Systems
Engineering and the School of Electrical and Computer Engineering (ECE).
He was born in Vietnam, where he got his Bachelor degree in
Automatic Control at Hanoi University of Science and Technology in 2008.
He obtained his Master and Ph.D. degrees both in ECE from the
University of Oklahoma in 2013 and the University of Illinois at
Urbana-Champaign in 2018, respectively. His research interests
lie at the intersection of control theory, optimization, distributed
algorithms, and applied probability, with the main applications in
machine learning, reinforcement learning, power networks, and
multi-agent systems.

Series: Combinatorics Seminar

The purpose of this talk is to explain the following result.
Let n > 2 be an integer. Let H be a 3-connected minor of a
3-connected graph G. If G is sufficiently large (relative to
n and the size of H) then G has a 3-connected minor obtained
from H by “adding” K_{3,n} or W_n.

Friday, November 2, 2018 - 15:05 ,
Location: Skiles 156 ,
Yian Yao ,
GT Math ,
Organizer: Jiaqi Yang

The
Shadowing lemma describes the behaviour of pseudo-orbits near a
hyperbolic invariant set. In this talk, I will present an analytic
proof of the shadowing lemma for
discrete flows. This is a work by K. R. Meyer and George R. Sell.